Field of the Invention
The invention relates generally to the field of determining subsurface geologic structures and formation composition (i.e., spatial distribution of one or more physical properties) by inversion processing of geophysical measurements. More particularly, the invention relates to methods for determining uncertainty in inversion results.
Background Art
In the present description of the Background of the Invention and in the Detailed Description which follows, references to the following documents are made:
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Alumbaugh, D. L., 2002, Linearized and nonlinear parameter variance estimation for two-dimensional electromagnetic induction inversion, Inverse Problems, 16, 1323-1341.
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Fernández-Álvarez, J. P., J. L. Fernández-Martínez, and C. O. Menéndez-Pérez, 2008, Feasibility analysis of the use of binary genetic algorithms as importance samplers application to a geoelectrical VES inverse problem, Mathematical Geosciences, 40, 375-408.
Fernández-Martínez, J. L., E. García-Gonzalo, J. P. F. Álvarez, H. A. Kuzma, and C. O. Menéndez-Pérez, 2010a, PSO: A powerful algorithm to solve geophysical inverse problems: Application to a 1D-DC Resistivity Case, Journal of Applied Geophysics, Accepted.
Fernández-Martínez, J. L., E. García-Gonzalo, and V. Naudet, 2010b, Particle swarm optimization applied to the solving and appraisal of the streaming potential inverse problem, Geophysics, Hydrogeophysics Special Issue, Accepted.
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Haario, H., E. Saksman, and J. Tamminen, 2001, An adaptive Metropolis algorithm, Bernoulli, 7, 223-242.
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Meju, M. A., and V. R. S. Hutton, 1992, Iterative most-squares inversion: application to magnetotelluric data, Geophys. J. Int., 108, 758-766.
Meju, M. A., 1994, Geophysical data analysis: understanding inverse problem theory and practice, Course Notes: Society of Exploration Geophysicists, Tulsa.
Meju, M. A., 2009, Regularized extremal bounds analysis (REBA): an approach to quantifying uncertainty in nonlinear geophysical inverse problems, Geophys. Res. Lett., 36, L03304.
Oldenburg, D. W., 1983, Funnel functions in linear and nonlinear appraisal, J. Geophys. Res., 88, 7387-7398.
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Sambridge, M., 1999, Geophysical inversion with a neighborhood algorithm-I. Searching a parameter space, Geophys. J. Int., 138, 479-494.
Sambridge, M., K. Gallagher, A. Jackson, and P. Rickwood, 2006, Trans-dimensional inverse problems, model comparison and the evidence, Geophys. J. Int., 167, 528-542, doi:10.1111/j.1365-246X.2006.03155.xScales, J. A., and L. Tenorio, 2001, Prior information and uncertainty in inverse problems, Geophysics, 66, 389-397.
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When solving geophysical problems, often the focus is on a complete solution to a particular inverse problem given a preferred inversion processing technique and any available knowledge of the geology (i.e., the structure and composition of the subsurface formations being evaluated). However, there is always the ancillary problem of quantifying how uncertain it is that the particular solution obtained is unique, or is even the best solution consistent with the actual geology (i.e., the actual spatial distribution of rock formations and corresponding physical properties in the subsurface). There are a number of reasons for uncertainty in inversion results, the most important of which are physical parameter measurement error, inversion solution non-uniqueness, density of the physical parameter measurements within a selected inversion volume (“data coverage”) and bandwidth limitation, and physical assumptions (e.g., isotropy) or approximations (numerical error). In the context of nonlinear inversion, the uncertainty problem is that of quantifying the variability in the model space supported by prior information and measured geophysical and/or petrophysical data. Because uncertainty is present in all geophysical inversion solutions, any geological interpretation made using inversion should include an estimate of the uncertainty. This is not typically the case, however. Rather, nonlinear inverse uncertainty remains one of the most significant unsolved problems in geophysical data interpretation, especially for large-scale inversion problems.
There are some methods known in the art for estimating inverse solution uncertainties (See Tarantola, 2005); however, these methods have been shown to be deficient for large-scale nonlinear inversion problems. As explained in Meju (2009), perhaps the most apparent distinction is between deterministic and stochastic methods. Deterministic methods seek to quantify inversion uncertainty based on least-squares inverse solutions and the computation of model resolution and covariance (e.g., Osypov et al., 2008; Zhang and Thurber, 2007) or by extremal solutions (e.g., Oldenburg, 1983; Meju, 2009), while stochastic methods seek to quantify uncertainty by presenting a problem in terms of random variables and processes and computing statistical moments of the resulting ensemble of solutions (e.g., Tarantola and Valette, 1982; Sambridge, 1999; Malinverno, 2002). Commonly, deterministic methods rely on linearized estimates of inverse model uncertainty, for example, about the last iteration of a nonlinear inversion, and thus, have limited relevance to actual nonlinear uncertainty (e.g., Meju, 1994; Alumbaugh, 2002). Stochastic uncertainty methods, which typically use random sampling schemes in parameter space, avoid burdensome inversions and account for nonlinearity but often come at the high computational cost of a very large number of forward solutions (Haario et al., 2001).
Other researchers have extended deterministic techniques or combined them with stochastic methods. Meju and Hutton (1992) presented an extension to linearized uncertainty estimation for magnetotelluric (MT) problems by using an iterative most-squares solution; however, due to its iterative extremizing of individual parameters, this method is practical only for small parameter spaces. Another approach has been to use the computational efficiency of deterministic inverse solutions and incorporate nonlinearity by probabilistic sampling (e.g., Materese, 1995; Alumbaugh and Newman, 2000; Alumbaugh, 2002). In essence, the foregoing hybrid method involves solving either a portion or the entire nonlinear inverse problem many times, while either the observations or prior model are treated as random variables. Such quasi-stochastic uncertainty method is able to account for at least a portion of the nonlinear uncertainty of geophysical inverse problems, but random sampling can be computationally inefficient and involves at least hundreds of inverse solutions (Alumbaugh, 2002) for only modest-sized problems.
The problem of uncertainty has a natural interpretation in a Bayesian framework (See Scales and Tenorio, 2001) and is very well connected to the use of sampling and a class of global optimization methods where the random search is directed using some fitness criteria for the estimates. Methods such as simulated annealing, genetic algorithm, particle swarm, and neighborhood algorithm belong to this category, and these can be useful for nonlinear problems (e.g., Sen and Stoffa, 1995; Sambridge, 1999; Fernández Alvarez et al., 2008; Fernández Martínez et al., 2010a,b). These stochastic methods avoid having to solve the large-scale inverse problem directly, account for problem nonlinearity, and produce estimates of uncertainty; however, they do not avoid having to sample the correspondingly massive multivariate posterior space (e.g., Haario et al., 2001). While this has limited the use of global optimization to nonlinear problems of modest size, recent work by Sambridge et al. (2006) suggests that extensions to somewhat larger parameterizations may be possible if parameter reduction is performed by optimization. Because most practical geophysical parameterizations consist of thousands to billions of unknowns, stochastic sampling of the entire model space is, at best, impractical. So, this begs the question: how can we reduce the computational burden of posterior sampling methods without limiting our uncertainty estimations to inaccurate linearizations?
We address this by presenting an alternative nonlinear scheme that infers uncertainty from sparse posterior sampling in bounded reduced-dimensional model spaces (Tompkins and Fernández Martínez, 2010). We adapt this method from Ganapathysubramanian and Zabaras (2007), who used it to solve the stochastic forward problem describing thermal diffusion through random heterogeneous media. The foregoing researchers showed that they could dramatically improve the efficiency of stochastic sampling if they combined model parameter reduction, parameter constraint mapping, and sparse deterministic sampling using a Smolyak scheme. Specifically, they computed model covariances from a statistical sampling of material properties (microstructures), and used Principal Component Analysis (PCA) to decorrelate and reduce their over-parameterized model domain by orders of magnitude. They then mapped parameter constraints, given statistical properties from the original model domain, to this reduced space, using a linear programming scheme. This was necessary, because they did not use the reduced base to restrict the values of parameters (only to restrict their spatial correlations). Ganapathysubramanian and Zabaras (2007) demonstrated that the bounded region defined in the reduced space was a “material” plane of equal probability and could be sampled to solve the forward stochastic thermal diffusion problem. While this method worked well for their forward problem, where the material property statistics are known, it is insufficient to solve the nonlinear inverse uncertainty problem. Here, we adapt this method to the geophysical uncertainty problem.